Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

The main objective of this paper is to relate the height and the number of generators of ideals in rings that are not necessarily Noetherian. As in [10, 11], we call an ideal I of a ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. This is a generalization of the notion of set theoretic complete intersection of ideals in Noetherian rings to rings that need not be Noetherian. In this work, we determine conditions on a ring R so that the prime ideals of R and also those of the polynomial rings R[X] over R are radically perfect. In many cases, it is shown that the condition of prime ideals of R or that of R[X] being radically perfect is equivalent to a form of the class group of R being torsion.

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Radical perfectness of prime ideals in certain integral domains

Chang¹,

Kim²

2017

J. Commut. Algebra

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For a UMT-domain D, we characterize when the polynomial ring D[X] is t-compactly packed and every prime t-ideal of D[X] is radically perfect. As a corollary, for a quasi-Prüfer domain D, we also characterize when every prime ideal of D[X] is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively. Introduction. LetD be an integral domain. In [15], Erdogdu introduced the notion of radical perfectness of ideals as follows: An ideal I of D is radically perfect if the height of I is equal to the infimum of the number of generators of ideals of D whose radical is equal to the radical of I. This generalizes the notion of a set-theoretic complete intersection to non-Noetherian rings [16, page 1802]. He then addressed the question of under which conditions all prime ideals of the polynomial ring D[X] over D containing a field of characteristic 0 are radically perfect. In this direction, it was shown [14, Theorem 2.1] that over a Noetherian domain D of Krull dimension 1 containing a field of characteristic 0, every prime ideal of D[X] is radically perfect if and only if D is a Dedekind domain with torsion ideal class group. In [14, Question 3.3], he also posed the open question:

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Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

Cited by 4 publications

References 7 publications

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

Commutative rings whose prime ideals are radically perfect

Radical perfectness of prime ideals in certain integral domains

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